On the Support of Minimizers of Causal Variational Principles

TL;DR

This paper introduces and analyzes a class of causal variational principles on compact manifolds, establishing properties of their minimizers and exploring explicit examples with connections to packing problems.

Contribution

It provides a general theoretical framework for minimizers of causal variational principles and detailed analyses for specific manifolds, including explicit solutions and bounds.

Findings

Minimizers are either fully timelike or have empty interior.

Explicit minimizer structures are characterized for circle, sphere, and flag manifolds.

Connections to packing problems and Tammes distribution are established.

Abstract

A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or it is singular in the sense that its interior is empty. In the examples of the circle, the sphere and certain flag manifolds, the general results are supplemented by a more detailed and explicit analysis of the minimizers. On the sphere, we get a connection to packing problems and the Tammes distribution. Moreover, the minimal action is estimated from above and below.